Julius'utility function is U(W)=ln(W). His current wealth is $5,000. He is now given a chance to buy a futures contract on Nickel that gives him 75% chance of winning $5,000, and 25% chance of losing $4,000. What is his, Julius' certainty equivalent for holding the futures contract?

Current Wealth = $5,000

Probability of Winning = 75%

Probability of Winning = 25%

Amount of Winning = $5,000

Amount of Loosing = $1,000

Wealth in Case of Winning = Current Wealth + Amount of Winning

Wealth in Case of Winning = 5,000 + 5,000

Wealth in Case of Winning = $10,000

Utility in Case of Winning = Log(Wealth in Case of Winning)

Utility in Case of Winning = Log(10,000)

Utility in Case of Winning = 9.21

Wealth in Case of Loosing = Current Wealth - Amount of Loosing

Wealth in Case of Loosing = 5,000 - 4,000

Wealth in Case of Loosing = $1,000

Utility in Case of Loosing = Log(Wealth in case of Loosing)

Utility in Case of Loosing = Log(1,000)

Utility in Case of Loosing = 6.91

Expected Utility Value = Utility in Case of Winning * Probability of Winning + Utility in Case of Loosing * Probability of Loosing

Expected Utility Value = 75% * 9.21 + 25% * 6.91

Expected Utility Value = 8.63

Since we want to have a certainity equivalent which would give the same utility as the expected in holding a future contract,

Utility of Certainity Equivalent of Holding Future Contract = Expected Utility Value

Log(Certainity Equivalent of Holding Future Contract) = Expected Utility Value

Certainity Equivalent of Holding Future Contract = e^{Expected Utility Value}

Certainity Equivalent of Holding Future Contract = e^8.63

Certainity Equivalent of Holding Future Contract = $5,623.41

Julius's Certainity Equivalent of Holding Future Contract is $5,623.41.

This effectively means that Julius would be indifferent between receiving the fixed sum of $5,623.41 for sure and the payoff like future contract although expected value would be different.